The model () can be regarded as a generalization of the model H with dynamic boundary conditions of Cahn–Hilliard type.
We will focus on the mathematical analysis of system () in the case \(L=0\)  and \(\beta >0\)  (see system (REF ) in Section ). In Theorem REF , we first prove the existence of global weak solutions to system (REF ) with regular (polynomial-like) potentials \(F\)  and \(G\)  in bounded Lipschitz domains (cf. (REF )–(REF ) below for the specific assumptions). Weak solutions satisfy the variational formulation of the problem (REF ) and a weak energy dissipation law (see (REF )). A weak solution is obtained as the limit of smooth solutions to suitably regularized problems based on a Faedo–Galerkin scheme, where appropriate compactness properties are derived on the basis of uniform and global estimates gained from the energy dissipation law (REF ).
However, an ad hoc construction of the solution ansatz is required since both \((\phi ,\psi )\)  and \((\mu ,\theta )\)  need to be approximated in the same finite-dimensional subspace in order to show the validity of (REF ) for such regularized solutions (cf. the change of variables (REF )). In comparison with the works in literature [1]}, [2]}, [3]}, our argument is rather simple and relies only on a one-level Faedo–Galerkin approximation. Furthermore, it is shown that \((\mu ,\theta )\in L^4(0,T; \mathcal {L}^2)\) , which is a novel regularity for NSCH systems (and even CH equations) with regular potentials (cf., for instance, [4]}, [5]}).
Next, in Theorem REF , we then establish the uniqueness of the weak solutions in the two dimensional case. This is the first uniqueness result in the literature for a NSCH system with dynamic boundary. It is important to underline that the nonlinear coupling in the NSCH system with dynamic boundary conditions () is much stronger than in the model H (). Roughly speaking, this is due to the elastic (surface) term \([\psi \nabla \theta ]_\tau \)  in (),
in addition to the classical capillarity (bulk) term \(-\varepsilon \textnormal {div}(\nabla \phi \otimes \nabla \phi )\)  (also referred to as Korteweg force).
More precisely, gradient terms arising from equivalent formulations of the Korteweg force (cf. (REF )) disappear in bulk integrals in view of the incompressibility constraint \(\textnormal {div}\, {v}=0\) .
Instead, surface integrals involving the product between a gradient function and \({v}\)  does not vanish since \(\textnormal {div}_\tau \, {v}\ne 0\)  on \(\Sigma \) . For this reason,
the term \([\psi \nabla \theta ]_\tau \)  cannot be handled as in [6]} through dual estimates.
Therefore, the appropriate functional to control the difference of two solutions corresponds to the total energy. This functional is capable of reproducing the same cancellation of the nonlinear terms as in the derivation of (REF ).
In order to rigorously justify this fact, two additional assumptions need to be imposed: the domain \(\Omega \)  has a \(C^3\) -boundary, which is needed to gain further spatial regularity of \((\phi ,\psi )\)  and to recover the relations of the bulk/surface chemical potentials \((\mu ,\theta )\)  almost everywhere; the relation between bulk and surface potentials \(F(s)=\beta G(s)\)  for all \(s \in \mathbb {R}\) , which enables us to apply a particular chain formula (see Appendix ).
